chaos.tex

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\begin{document}

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% -------------- TITLE PAGE --------------
\begin{titlingpage}
    \centering

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        \Huge
        \parbox{10cm}{\begin{center}{\scshape \textbf{Chaos in Topological Dynamical Systems}}\end{center}}


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        \LARGE
        {\scshape \textbf{Fraser Robert Love}}
             
        \vspace{0.75cm}
             
        \large

        School of Mathematics and Statistics\\
        University of St Andrews\\

        \vspace{3cm}

        A dissertation submitted for the degree of\\
        \textit{BSc (Hons) Mathematics}\\

        \vfill

        {\large \today\par}
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\vspace*{2.5cm}

\noindent\textit{I certify that this project report has been written by me, is a record of work carried out by me, and is essentially different from work undertaken for any other purpose or assessment.}

\begin{center}
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\vspace{1cm}

\begin{abstract}
    \noindent A topological dynamical system is comprised of a continuous mapping acting on a compact metric space. This project studies the complex, chaotic behaviour that can arise in these systems. Using the extra condition of compactness present in these systems, proves beneficial in analysis of chaos and the behaviour of these systems as the underlying mapping is iterated \emph{ad infinitum}. Various definitions of chaos will be examined, namely, Devaney chaos, Li-Yorke chaos and topological chaos. These definitions encompass aspects of indecomposability, repetitiveness and unpredictability; which when combined give a natural interpretation of chaos. This project shall study how these definitions specifically apply to topological dynamical systems on the interval, the unit circle, in sequence space and on compact countable sets. Numerous important topological properties of chaos will be introduced; such as topological transitivity, sensitive dependence, dense periodic points, scrambled sets, Li-Yorke pairs and positive topological entropy. Foundational tools from symbolic dynamics will be combined with topological conjugacy, to transfer these topological properties between systems. Finally, this text shall conclude by characterising various chaotic systems and comparing the definitions of chaos.
\end{abstract}

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\renewcommand{\abstractname}{Acknowledgements}
\begin{abstract}
    \noindent I would like to express my gratitude to Prof.\ Mike Todd for his invaluable guidance and support throughout my time researching this honours project.
    
\end{abstract}

\newpage
\tableofcontents

\chapter{An Introduction to Topological Dynamics} \label{chap:introduction}
\input{chapters/an_introduction_to_topological_dynamics}

\chapter{Topological and Symbolic Relationships} \label{chap:conjugacy-symbol-dynamics}
\input{chapters/topological_and_symbolic_relationships}

\chapter{Topological Characteristics and Definitions of Chaos} \label{chap:defining-chaos}
\input{chapters/defining_chaos}

\chapter{Conclusion} \label{chap:conclusion}
\input{chapters/conclusion}

\bibliography{chaos}

\end{document}